As far as axiomatic morality goes, I've tried very diligently to develop moral axioms in order to construct a rigorous moral system, but it's proven to be extremely difficult, if not impossible, because edge cases always seem to appear under some set of conditions. Even the axiom of, "murder is immoral/morally bad/morally negative," can have a plausible edge case where it may be justifiable to commit murder. Why? Because it could, hypothetically, save lives in the immediate. An example could be a madman who orders you to kill an innocent person or they will detonate a nuclear bomb killing millions. Variations of this hypothetical are typical in discussions of the contrast between utilitarianism and deontology.
Metaethically, I don't think ethics are amenable to axiomatization. If it is, in fact, possible, it may be a problem of complexity. The space of possible moral agentic decisions (actions a thing with agency, such as a person, can take) is a subset of the space of all possible agentic decisions (because some decisions have a null moral value).